Please help with a matrix multiplicative inverse problem

Use the elementary row transformations to find the multiplicative inverse (if one exists) of matrix A. (Objective 4. If an answer does not exist, enter DNE.)
A=
[6 1 -6
13 -4 1
7 -5 7]

I just started learning the concept, this problem is too hard to me no matter how many times I tried, please help out, thanks

Row reduce [A : I] to [I : A⁻¹], if the inverse exists.

Start with
[6 1 -6: 1 0 0]
[13 -4 1: 0 1 0]
[7 -5 7: 0 0 1].

R3 - R1 --> R1:
[1 -6 13: -1 0 1]
[13 -4 1: 0 1 0]
[7 -5 7: 0 0 1].

13R1 - R2 --> R2 and 7R1 - R3 --> R3:
[1 -6 13: -1 0 1]
[0 -74 168: -13 -1 0]
[0 -37 84: -7 0 6].

R2/2 --> R2:
[1 -6 13: -1 0 1]
[0 -37 84: -13/2 -1/2 0]
[0 -37 84: -7 0 6].

R2 - R3 --> R3:
[1 -6 13: -1 0 1]
[0 -37 84: -13/2 -1/2 0]
[0 0 0: 1/2 -1/2 -6].

Since we have a row of zeros on the left half row, the inverse does not exist.

I hope this helps!